Asymptotic analysis and optimal control of an integro-differential system modelling healthy and cancer cells exposed to chemotherapy

Abstract

We consider a system of two coupled integro-differential equations modelling populations of healthy and cancer cells under therapy. Both populations are structured by a phenotypic variable, representing their level of resistance to the treatment. We analyse the asymptotic behaviour of the model under constant infusion of drugs. By designing an appropriate Lyapunov function, we prove that both densities converge to Dirac masses. We then define an optimal control problem, by considering all possible infusion protocols and minimising the number of cancer cells over a prescribed time frame. We provide a quasi-optimal strategy and prove that it solves this problem for large final times. For this modelling framework, we illustrate our results with numerical simulations, and compare our optimal strategy with periodic treatment schedules.

Why Is Evolution Important in Cancer and What Mathematics Should Be Used to Treat Cancer? Focus on Drug Resistance

“…Under hypothesis (4.1), it is proved in [Pouchol et al 2016] 32 that the behaviour of (1) is again convergence and concentration, where the asymptotic values of ρ H , ρ C and the sets on which n H , n C concentrate can also be explicitly computed.…”

“…Optimal control methods (reviewed in [Trélat 2005] 36 ) applied to models of cancer therapeutics using systems of ordinary differential equations [Carrère 2017; Schättler 2006, 2014] 5,21,22 or of partial differential equations [Pouchol et al 2016] 32 are the appropriate tool to theoretically optimise cancer therapeutics, in particular by taking into account the inevitable emergence of drug resistance in cancer cell populations.…”

“…In [Pouchol et al 2016] 32 , the previously defined optimal control problem is analysed both numerically and theoretically. As the time T increases, it is found that the optimal control strategy becomes increasingly close to a two-phase strategy.…”

Bifurcation analysis for a fractional‐order chemotherapy model with two different delays

“…For example, Lai and Friedman 5 investigated the cancer treatment with a combination of anti-VEGF (bevacizumab) and a chemotherapy drug; Liu and Liu 6 studied the positivity and boundedness of solutions, local and global stability of the equilibrium points of a tumor-immune model considering targeted chemotherapy. Schättler et al 7 discussed the dynamical behavior of a metronomic chemotherapy model; Pouchol et al 8 focused on the asymptotic analysis and optimal control of integro-differential equations modeling populations of healthy and cancer cells under chemotherapy. In details, one can see previous studies.…”

Modeling continuous levels of resistance to multidrug therapy in cancer

“…The biological mechanisms responsible for the emergence of drug resistance and its propagation often involve a multifactorial and complex process of genetic and epigenetic alternations [1][2][3], that arise through a series of genetic and non-genetic changes [4][5][6][7]. Such changes can be due to drug administration (drug induced resistance) [8,9], or they can emerge independent of therapy due to intrinsic mechanisms. Cancer cells may develop simultaneous resistance to structurally and mechanistically unrelated drugs, leading to multidrug resistance (MDR) [1,2,10].…”